|x - 1| < d means, that the distance of x to 1 is no more than d. Now what is the distance of x to -1? Well, if x is "to the left" of 1, then it is definitely closer to -1 than d. For example, if x is at the maximal distance from 1, namely 1 - d, then it has distance 1 - (1 - d) = d to -1.
If x is on the other side of 1, its distance to -1 is equal to two units plus the distance to 1.

If you use the idea that you can measure the distance of x to -1 by going "to 1" first, you hopefully see how the triangle inequality comes in useful.